# EEE 431 Computational Methods in Electrodynamics - PowerPoint PPT Presentation

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EEE 431 Computational Methods in Electrodynamics. Lecture 3 By Dr. Rasime Uyguroglu. Energy and Power. We would like to derive equations governing EM energy and power. Starting with Maxwell’s equation’s:. Energy and Power (Cont.). Apply H. to the first equation and E. to the second:.

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EEE 431 Computational Methods in Electrodynamics

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## EEE 431Computational Methods in Electrodynamics

Lecture 3

By

Dr. Rasime Uyguroglu

### Energy and Power

• We would like to derive equations governing EM energy and power.

• Starting with Maxwell’s equation’s:

### Energy and Power (Cont.)

• Apply H. to the first equation and E. to the second:

• Subtracting:

• Since,

### Energy and Power (Cont.)

• Integration over the volume of interest:

### Energy and Power (Cont.)

• Applying the divergence theorem:

### Energy and Power (Cont.)

• Explanation of different terms:

• Poynting Vector in

• The power flowing out of the surface S (W).

### Energy and Power (Cont.)

• Dissipated Power (W)

• Supplied Power (W)

### Energy and Power

• Magnetic power (W)

• Magnetic Energy.

### Energy and Power (Cont.)

• Electric power (W)

• electric energy.

### Energy and Power (Cont.)

• Conservation of EM Energy

### Classification of EM Problems

• 1) The solution region of the problem,

• 2) The nature of the equation describing the problem,

• 3) The associated boundary conditions.

### 1) Classification of Solution Regions:

• Closed region, bounded, or open region, unbounded. i.e Wave propagation in a waveguide is a closed region problem where radiation from a dipole antenna is an open region problem.

• A problem also is classified in terms of the electrical, constitutive properties. We shall be concerned with simple materials here.

### 2)Classification of differential Equations

• Most EM problems can be written as:

• L: Operator (integral, differential, integrodifferential)

• : Excitation or source

• : Unknown function.

### Classification of Differential Equations (Cont.)

• Example: Poisson’s Equation in differential form .

### Classification of Differential Equations (Cont.):

• In integral form, the Poisson’s equation is of the form:

### Classification of Differential Equations (Cont.):

• EM problems satisfy second order partial differential equations (PDE).

• i.e. Wave equation, Laplace’s equation.

### Classification of Differential Equations (Cont.):

• In general, a two dimensional second order PDE:

• If PDE is homogeneous.

• If PDE is inhomogeneous.

### Classification of Differential Equations (Cont.):

• A PDE in general can have both:

• 1) Initial values (Transient Equations)

• 2) Boundary Values (Steady state equations)

### Classification of Differential Equations (Cont.):

• The L operator is now:

### Classification of Differential Equations (Cont.):

• Examples:

• Elliptic PDE, Poisson’s and Laplace’s Equations:

### Classification of Differential Equations (Cont.):

• For both cases a=c=1,b=0.

• An elliptic PDE usually models the closed region problems.

### Classification of Differential Equations (Cont.):

• Hyperbolic PDE’s, the Wave Equation in one dimension:

• Propagation Problems (Open region problems)

### Classification of Differential Equations (Cont.):

• Parabolic PDE, Heat Equation in one dimension.

• Open region problem.

### Classification of Differential Equations (Cont.):

• The type of problem represented by:

• Such problems are called deterministic.

• Nondeterministic (eigenvalue) problem is represented by:

• Eigenproblems: Waveguide problems, where eigenvalues corresponds to cutoff frequencies.

### 3) Classification of Boundary Conditions:

• What is the problem?

• Find which satisfies within a solution region R.

• must satisfy certain conditions on Surface S, the boundary of R.

• These boundary conditions are Dirichlet and Neumann types.

### Classification of Boundary Conditions (Cont.):

• 1) Dirichlet B.C.:

• vanishes on S.

• 2) Neumann B.C.:

• i.e. the normal derivative of vanishes on S.

• Mixed B.C. exits.